A self-exciting spatio-temporal model with a smooth space-time-varying productivity parameter

The self-exciting spatio-temporal point process model is a fundamental tool for studying recurrent events in fields such as economics, criminology, and seismology. Existing models often assume that the productivity parameter, which measures the rate of triggered events, is constant in space and time. This assumption is often unrealistic, as it may not capture the complexity of some real-world phenomena. In this paper, we propose a new self-exciting model that relaxes this assumption by allowing the productivity parameter to vary smoothly in both space and time. Through simulation experiments, we demonstrate that our model can effectively recover the underlying pattern of excitation. Furthermore, we apply the proposed framework to a crime dataset, showing its ability to identify spatial and temporal heterogeneity in event dynamics. This approach offers a more realistic method for modeling spatio-temporal patterns, with significant potential for the development of surveillance and prevention tools in a range of applications.